Some estimates of Schrödinger type operators on variable Lebesgue and Hardy spaces

Abstract

In this article, the authors consider the Schrödinger type operator \(L:=-\mathrm{div}(A\nabla )+V\) on \({\mathbb {R}}^n\) with \(n\ge 3\), where the matrix A satisfies uniformly elliptic condition and the nonnegative potential V belongs to the reverse Hölder class \(RH_q({\mathbb {R}}^n)\) with \(q\in (n/2,\,\infty )\). Let \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\,\infty )\) be a variable exponent function satisfying the globally \(\log \)-Hölder continuous condition. When \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (1,\,\infty )\), the authors prove that the operators \(VL^{-1}\), \(V^{1/2}\nabla L^{-1}\) and \(\nabla ^2L^{-1}\) are bounded on variable Lebesgue space \(L^{p(\cdot )}({\mathbb {R}}^n)\). When \(p(\cdot ):\ {\mathbb {R}}^n\rightarrow (0,\,1]\), the authors introduce the variable Hardy space \(H_L^{p(\cdot )}({\mathbb {R}}^n)\), associated to L, and show that \(VL^{-1}\), \(V^{1/2}\nabla L^{-1}\) and \(\nabla ^2L^{-1}\) are bounded from \(H_L^{p(\cdot )}({\mathbb {R}}^n)\) to \(L^{p(\cdot )}({\mathbb {R}}^n)\).